On the endpoint regularity of discrete maximal operators

Given a discrete function $f:\Z^d \to \R$ we consider the maximal operator $Mf(\vec{n}) = \sup_{r\geq0} \frac{1}{N(r)} \sum_{\vec{m} \in \barΩ_r} \big|f(\vec{n} + \vec{m})\big|,$ where $\big\{\barΩ_r\big\}_{r \geq 0}$ are dilations of a convex set $Ω$ (open, bounded and with Lipschitz boudary) containing the origin and $N(r)$ is the number of lattice points inside $\barΩ_r$. We prove here that the operator $f \mapsto \nabla M f$ is bounded and continuous from $l^1(\Z^d)$ to $l^1(\Z^d)$. We also prove the same result for the non-centered version of this discrete maximal operator.

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